Euler equations
Monika Polner Bolyai Institute,
University of Szeged, Aradi v´ertan´uk tere 1, 6720 Szeged, Hungary polner@math.u-szeged.hu
J.J.W. van der Vegt Department of Applied Mathematics,
University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands j.j.w.vandervegt@utwente.nl
Using the Hodge decomposition on bounded domains the compressible Euler equations of gas dynamics are reformulated using a density weighted vorticity and dilatation as primary variables, together with the entropy and density. This formulation is an extension to compressible flows of the well-known vorticity-stream function formulation of the incompressible Euler equations. The Hamiltonian and associated Poisson bracket for this new formulation of the compressible Euler equations are derived and extensive use is made of differential forms to highlight the mathematical structure of the equations. In order to deal with domains with boundaries also the Stokes-Dirac structure and the port-Hamiltonian formulation of the Euler equations in density weighted vorticity and dilatation variables are obtained.
Keywords: Compressible Euler equations; Hamiltonian formulation; de Rham complex; Hodge decomposition; Stokes-Dirac structures, vorticity, dilatation.
AMS Subject Classification: 37K05, 58A14, 58J10, 35Q31, 76N15, 93C20, 65N30.
1. Introduction
The dynamics of an inviscid compressible gas is described by the compressible Eu-ler equations and equation of state. The compressible EuEu-ler equations have been extensively used to model many different types of compressible flows, since in many applications the effects of viscosity are small or can be neglected. This has moti-vated over the years extensive theoretical and numerical studies of the compressible Euler equations. The Euler equations for a compressible, inviscid and non-isentropic gas in a domain Ω ⊆ R3are defined as
ρt= −∇ · (ρu), (1.1) ut= −u · ∇u − 1 ρ∇p, (1.2) st= −u · ∇s, (1.3) 1
with u = u(x, t) ∈ R3
the fluid velocity, ρ = ρ(x, t) ∈ R+ the mass density and s(x, t) ∈ R the entropy of the fluid, which is conserved along streamlines. The spatial coordinates are x ∈ Ω and time t and the subscript means differentiation with respect to time. The pressure p(x, t) is given by an equation of state
p = ρ2∂U
∂ρ(ρ, s), (1.4)
where U (ρ, s) is the internal energy function that depends on the density ρ and the entropy s of the fluid. The compressible Euler equations have a rich mathematical structure15 and can be represented as an infinite dimensional Hamiltonian system 12,13. Depending on the field of interest, various types of variables have been used to define the Euler equations, e.g. conservative, primitive and entropy variables 15. The conservative variable formulation is for instance a good starting point for numerical discretizations that can capture flow discontinuities 10, such as shocks and contact waves, whereas the primitive and entropy variables are frequently used in theoretical studies.
In many flows vorticity is, however, the primary variable of interest. Histori-cally, the Kelvin circulation theorem and Helmholtz theorems on vortex filaments have played an important role in describing incompressible flows, in particular the importance of vortical structures. This has motivated the use of vorticity as pri-mary variable in theoretical studies of incompressible flows, see e.g. 1,12, and the development of vortex methods to compute incompressible vortex dominated flows 7.
The use of vorticity as primary variable is, however, not very common for com-pressible flows. This is partly due to the fact that the equations describing the evolution of vorticity in a compressible flow are considerably more complicated than those for incompressible flows. Nevertheless, vorticity is also very important in many compressible flows. A better insight into the role of vorticity, and also di-latation to account for compressibility effects, is not only of theoretical importance, but also relevant for the development of numerical discretizations that can compute these quantities with high accuracy.
In this article we will present a vorticity-dilatation formulation of the compress-ible Euler equations. Special attention will be given to the Hamiltonian formulation of the compressible Euler equations in terms of the density weighted vorticity and dilatation variables on domains with boundaries. This formulation is an extension to compressible flows of the well-known vorticity-stream function formulation of the incompressible Euler equations1,12. An important theoretical tool in this analysis is the Hodge decomposition on bounded domains 18. Since bounded domains are crucial in many applications we also consider the Stokes-Dirac structure of the com-pressible Euler equations. This results in a port-Hamiltonian formulation17 of the compressible Euler equations in terms of the vorticity-dilatation variables, which clearly identifies the flows and efforts entering and leaving the domain. An impor-tant feature of our presentation is that we extensively use the language of
differen-tial forms. Apart from being a natural way to describe the underlying mathematical structure it is also important for our long term objective, viz. the derivation of finite element discretizations that preserve the mathematical structure as much as possi-ble. A nice way to achieve this is by using discrete differential forms and exterior calculus, as highlighted in2,3,19.
The outline of this article is as follows. In the introductory Section 2 we sum-marize the main techniques that we will use in our analysis. A crucial element is the use of the Hodge decomposition on bounded domains, which we briefly discuss in Section 2.2. This analysis is based on the concept of Hilbert complexes, which we summarize in Section 2.1. The Hodge Laplacian problem is discussed in Section 2.3. Here we show how to deal with inhomogeneous boundary conditions, which is of great importance for our applications. These results will be used in Section 3 to define via the Hodge decomposition a new set of variables, viz., the density weighted vorticity and dilatation, and to formulate the Euler equations in terms of these new variables. Section 4 deals with the Hamiltonian formulation of the Euler equations using the density weighted vorticity and dilatation, together with the density and entropy, as primary variables. The Poisson bracket for the Euler equations in these variables is derived in Section 5. In order to account for bounded domains we ex-tend the results obtained for the Hamiltonian formulation in Sections 4 and 5 to the port-Hamiltonian framework in Section 6. First, we extend in Section 6.1 the Stokes-Dirac structure for the isentropic compressible Euler equations presented in16 to the non-isentropic Euler equations. Next, we derive the Stokes-Dirac struc-ture for the compressible Euler equations in the vorticity-dilatation formulation in Section 6.3 and use this in Section 6.5 to obtain a port-Hamiltonian formulation of the compressible Euler equations in vorticity-dilatation variables. Finally, in Section 7 we finish with some conclusions.
2. Preliminaries
This preliminary section is devoted to summarize the main concepts and techniques that we use throughout this paper in our analysis.
2.1. Review of Hilbert complexes
In this section we discuss the abstract framework of Hilbert complexes, which is the basis of the exterior calculus in Arnold, Falk and Winther 3 and to which we refer for a detailed presentation. We also refer to Br¨uning and Lesch 6 for a functional analytic treatment of Hilbert complexes.
Definition 2.1. A Hilbert complex (W, d) consists of a sequence of Hilbert spaces Wk, along with closed, densely-defined linear operators dk : Wk → Wk+1, possibly unbounded, such that the range of dk is contained in the domain of dk+1 and dk+1◦ dk = 0 for each k.
A Hilbert complex is bounded if, for each k, dk is a bounded linear operator from Wk to Wk+1 and it is closed if for each k, the range of dk is closed in Wk+1. Definition 2.2. Given a Hilbert complex (W, d), a domain complex (V, d) consists of domains D(dk) = Vk ⊂ Wk, endowed with the graph inner product
hu, viVk= hu, viWk+dku, dkv
Wk+1.
Remark 2.1. Since dkis a closed map, each Vk is closed with respect to the norm induced by the graph inner product. From the Closed Graph Theorem, it follows that dk is a bounded operator from Vk to Vk+1. Hence, (V, d) is a bounded Hilbert complex. The domain complex is closed if and only if the original complex (W, d) is.
Definition 2.3. Given a Hilbert complex (W, d), the space of k-cocycles is the null space Zk = ker dk, the space of k-coboundaries is the image Bk = dk−1Vk−1, the kth harmonic space is the intersection Hk = Zk∩ Bk⊥W, and the kth reduced
cohomology space is the quotient Zk/Bk. When Bk is closed, Zk/Bk is called the kth cohomology space.
Remark 2.2. The harmonic space Hk is isomorphic to the reduced cohomology space Zk/Bk. For a closed complex, this is identical to the homology space Zk/Bk, since Bk is closed for each k.
Definition 2.4. Given a Hilbert complex (W, d), the dual complex (W∗, d∗) con-sists of the spaces Wk∗ = Wk, and adjoint operators d∗k = (dk−1)∗ : Vk∗ ⊂ W∗
k → Vk−1∗ ⊂ W∗
k−1. The domain of d∗k is denoted by Vk∗, which is dense in Wk.
Definition 2.5. We can define the k-cycles Z∗k = ker d∗k = Bk⊥W and k-boundaries
B∗k = d∗k+1Vk∗.
2.2. The L2-de Rham complex and Hodge decomposition
The basic example of a Hilbert complex is the L2-de Rham complex of differential forms. Let Ω ⊆ Rn be an n-dimensional oriented manifold with Lipschitz boundary ∂Ω, representing the space of spatial variables. Assume that there is a Riemannian metric , on Ω. We denote by Λk(Ω) the space of smooth differential k-forms on Ω, d is the exterior derivative operator, taking differential k-forms on the do-main Ω to differential (k + 1)-forms, δ represents the coderivative operator and ? the Hodge star operator associated to the Riemannian metric , . The opera-tions grad, curl, div, ×, · from vector analysis can be identified with operaopera-tions on differential forms, see e.g.9.
We can define the L2-inner product of any two differential k-forms on Ω as hω, ηiL2Λk = Z Ω ω ∧ ?η = Z Ω ω, η vΩ= Z Ω ?(ω ∧ ?η)vΩ, (2.1)
where vΩis the Riemannian volume form. The Hilbert space L2Λk(Ω) is the space of differential k-forms for which kωkL2Λk=phω, ωiL2Λk< ∞. When Ω is omitted
from L2Λkin the inner product (2.1), then the integral is always over Ω. The exterior derivative d = dk may be viewed as an unbounded operator from L2Λk to L2Λk+1. Its domain, denoted by HΛk(Ω), is the space of differential forms in L2Λk(Ω) with the weak derivative in L2Λk+1(Ω), that is
D(d) = HΛk(Ω) = {ω ∈ L2Λk(Ω) | dω ∈ L2Λk+1(Ω)}, which is a Hilbert space with the inner product
hω, ηiHΛk = hω, ηiL2Λk+ hdω, dηiL2Λk+1.
For an oriented Riemannian manifold Ω ⊆ R3, the L2 de Rham complex is 0 → L2Λ0(Ω)−→ Ld 2Λ1(Ω) d
−→ L2Λ2(Ω) d
−→ L2Λ3(Ω) → 0. (2.2) Note that d is a bounded map from HΛk(Ω) to L2Λk+1(Ω) and D(d) = HΛk(Ω) is densely-defined in L2Λk(Ω). Since HΛk(Ω) is complete with the graph norm, d is a closed operator (equivalent statement to the Closed Graph Theorem). Thus, the L2 de Rham domain complex for Ω ⊆ R3 is
0 → HΛ0(Ω)−→ HΛd 1(Ω)−→ HΛd 2(Ω)−→ HΛd 3(Ω) → 0. (2.3) The coderivative operator δ : L2Λk(Ω) 7→ L2Λk−1(Ω) is defined as
δω = (−1)n(k+1)+1? d ? ω, ω ∈ L2Λk(Ω). (2.4) Since we assumed that Ω has Lipschitz boundary, the trace theorem holds and the trace operator tr = tr∂Ω maps HΛk(Ω) boundedly into the Sobolev space H−1/2Λk(∂Ω). Moreover, the trace operator extends to a bounded surjection from H1Λk(Ω) onto H1/2Λ(∂Ω), see2. We denote the space HΛk(Ω) with vanishing trace as
◦
H Λk(Ω) = {ω ∈ HΛk(Ω) | tr ω = 0}. (2.5) In analogy with HΛk(Ω), we can define the space
H∗Λk(Ω) =ω ∈ L2Λk(Ω) | δω ∈ L2Λk−1(Ω) . (2.6) Since H∗Λk(Ω) = ?HΛn−k(Ω), for ω ∈ H∗Λk(Ω), the quantity tr(?ω) is well de-fined, and we can define
◦
H∗Λk(Ω) = ? ◦
H Λn−k(Ω) = {ω ∈ H∗Λk(Ω) | tr(?ω) = 0}. (2.7) The adjoint d∗= d∗k of dk−1 has domain D(d∗) =H◦∗Λk(Ω) and coincides with the operator δ defined in (2.4), (see3). Hence, the dual complex of (2.3) is
0 ← ◦ H∗Λ0(Ω)←−δ ◦ H∗Λ1(Ω)←−δ ◦ H∗Λ2(Ω)←−δ ◦ H∗Λ3(Ω) ← 0. (2.8)
The integration by parts formula also holds hdω, ηi = hω, δηi + Z ∂Ω tr ω ∧ tr(?η), ω ∈ Λk−1(Ω), η ∈ Λk(Ω), (2.9) and we have hdω, ηi = hω, δηi , ω ∈ HΛk−1(Ω), η ∈ ◦ H∗Λk(Ω). (2.10) Since the L2-de Rham complexes (2.2) and (2.3) are closed Hilbert complexes, the Hodge decomposition of L2Λk and HΛk are:
L2Λk = Bk⊕ Hk⊕B◦∗
k, (2.11)
HΛk = Bk⊕ Hk⊕ Zk⊥, (2.12)
whereB◦∗k= {δω | ω ∈H◦∗Λk+1(Ω)}, and Zk⊥= HΛk∩B◦∗
k denotes the orthogonal complement of Zk in HΛk. The space of harmonic forms, both for the original complex and the dual complex, is
Hk= {ω ∈ HΛk(Ω) ∩ ◦
H∗Λk(Ω) | dω = 0, δω = 0}. (2.13) Problems with essential boundary conditions are important for applications. This is why we briefly review the de Rham complex with essential boundary conditions. Take as domain of the exterior derivative dk the space H Λ◦ k(Ω). The de Rham complex with homogeneous boundary conditions on Ω ⊂ R3 is
0 → ◦
H Λ0(Ω)−→d H Λ◦ 1(Ω)−→d H Λ◦ 2(Ω)−→d H Λ◦ 3(Ω) → 0. (2.14) From (2.9), we obtain that
hdω, ηi = hω, δηi , ω ∈ ◦
H Λk−1(Ω), η ∈ H∗Λk(Ω). (2.15) Hence, the adjoint d∗ of the exterior derivative with domain H Λ◦ k(Ω) has domain H∗Λk(Ω) and coincides with the operator δ. Finally, the second Hodge decomposi-tion of L2Λk and of ◦ H Λk follow immediately: L2Λk =B◦k⊕ ◦ Hk⊕ B∗ k, (2.16) ◦ H Λk =B◦k⊕H◦k⊕ Zk⊥, (2.17) whereB◦k= d ◦ H Λk−1(Ω), Zk⊥=H Λ◦ k∩ B∗
k and the space of harmonic forms is ◦
Hk = {ω ∈ ◦
H Λk(Ω) ∩ H∗Λk(Ω) | dω = 0, δω = 0}. (2.18) 2.3. The Hodge Laplacian problem
In this section we first review the Hodge Laplacian problem with homogeneous natural and essential boundary conditions. The main result of this section is to show how to deal with inhomogeneous boundary conditions, which are crucial for applications.
2.3.1. The Hodge Laplacian problem with homogeneous natural boundary conditions
Given the Hilbert complex (2.3) and its dual complex (2.8), the Hodge Laplacian operator applied to a k-form is an unbounded operator Lk = dk−1d∗k+ d∗k+1d
k : D(Lk) ⊂ L2Λk→ L2Λk, with domain (see3)
D(Lk) = {u ∈ HΛk∩ ◦
H∗Λk | dku ∈ ◦
H∗Λk+1, d∗ku ∈ HΛk−1}. (2.19) In the following, we will not use the subscripts and superscripts k of the operators when they are clear from the context and use d∗= δ.
For any f ∈ L2Λk, there exists a unique solution u = K
kf ∈ D(Lk) satisfying
Lku = f (mod H), u ⊥ H, (2.20)
with Kk the solution operator, see3. The solution u satisfies the Hodge Laplacian (homogeneous) boundary value problem
(dδ + δd)u = f − PHf in Ω, tr(?u) = 0, tr(?du) = 0 on ∂Ω, (2.21) where PHf is the orthogonal projection of f into H, with the condition u ⊥ H required for uniqueness of the solution. The boundary conditions in (2.21) are both natural in the mixed variational formulation of the Hodge Laplacian problem, as discussed in 3.
The Hodge Laplacian problem is closely related to the Hodge decomposition in the following way. Since dδu ∈ Bkand δdu ∈ B∗
k, the differential equation in (2.21), or equivalently f = dδu + δdu + α, α ∈ Hk, is exactly the Hodge decomposition of f ∈ L2Λk(Ω). If we restrict f to an element of Bk or B∗k, we obtain two problems that are important for our applications (see also 3).
The B problem. If f ∈ Bk, then u = K
kf satisfies dδu = f, u ⊥ ◦ Z∗k, where ◦ Z∗ k = {ω ∈ ◦
H∗Λk(Ω) | δω = 0}. It also follows that the solution u ∈ Bk. To see this, consider u ∈ D(Lk) and the Hodge decomposition u = uB+ uH+ u⊥, where uB∈ Bk, uH∈ Hk and u⊥∈ B∗k∩ HΛ
k. Then, Lku = dδuB+ δdu⊥= f.
If f ∈ Bk, then u = uB, hence u ∈ Bk. Then, duB = 0 since d2 = 0, and since Bk=
◦
Z∗k⊥, it follows that u ⊥ ◦
Z∗k is also satisfied and u solves uniquely the Hodge Laplacian boundary value problem, see3,
dδu = f, du = 0, u ⊥ ◦
Z∗k in Ω (2.22)
tr(?u) = 0, on ∂Ω. (2.23)
The B∗ problem. If f ∈ B∗k, then u = Kkf satisfies δdu = f, with u ⊥ Zk. Similarly as for the B problem, it can be shown that the solution u ∈ B∗k. Consider u ∈ D(Lk) and the Hodge decomposition u = uB+ uH+ u⊥. Then,
If f ∈ B∗k, then u = u⊥, hence u ∈ B∗k. Therefore, δu⊥ = 0 and u ⊥ Zk. Conse-quently, u solves uniquely the Hodge Laplacian boundary value problem
δdu = f, δu = 0, u ⊥ Zk in Ω (2.24)
tr(?du) = 0, on ∂Ω. (2.25)
2.3.2. The Hodge Laplacian problem with homogeneous essential boundary conditions
Considering now the Hilbert complex with (homogeneous) boundary conditions (2.14) and its dual complex, the Hodge Laplacian problem with essential boundary conditions is
Lku = dδu + δdu = f (mod ◦
H), in Ω (2.26)
tr(δu) = 0, tr u = 0 on ∂Ω, (2.27) with the condition u ⊥H, which has a unique solution, u = K◦ kf. Both boundary conditions in (2.27) are essential in the mixed variational formulation of the Hodge Laplacian problem (see 3). Here the domain of the Laplacian is
D(Lk) = {u ∈ ◦
H Λk∩ H∗Λk| du ∈ H∗Λk+1, δu ∈H Λ◦ k−1}. (2.28) We can briefly formulate the B and B∗ problems as follows.
The B problem. If f ∈B◦k, then then u = Kkf ∈ ◦
Bk satisfies dδu = f, u ⊥ Z∗k. Then u solves uniquely the B problem
dδu = f, du = 0, u ⊥ Z∗k in Ω (2.29)
tr(δu) = 0, on ∂Ω. (2.30)
The B∗ problem. If f ∈ B∗k, then u = Kkf satisfies δdu = f, with u ⊥ Zk. Similarly, u solves uniquely
δdu = f, δu = 0, u ⊥ Zk in Ω (2.31)
tr(u) = 0, on ∂Ω. (2.32)
The next section shows how to transform the inhomogeneous Hodge Laplacian boundary value problem into a homogeneous one.
2.3.3. The Hodge Laplacian with inhomogeneous essential boundary conditions
Consider the case when the essential boundary conditions are inhomogeneous, that is, the boundary value problem
Lku = dδu + δdu = f in Ω (2.33)
with the condition hf, αi = Z ∂Ω rN∧ tr(?α), ∀α ∈ ◦ Hk. (2.35)
Here the domain of the Hodge Laplacian operator is DD k =u ∈ HΛ k∩ H∗Λk | du ∈ H∗Λk+1, δu ∈ HΛk−1, tr u = rb∈ H1/2Λk(∂Ω), tr(δu) = rN ∈ H1/2Λk−1(∂Ω) o . (2.36) This problem has a solution, which is unique up to a harmonic form α ∈ H◦k. Following the idea of Schwarz in 18, the inhomogeneous boundary value problem can be transformed to a homogeneous problem in the following way. For a given rb∈ H1/2Λk(∂Ω), using a bounded, linear trace lifting operator (see5,2,18), we can find η ∈ H1Λk(Ω), such that tr η = rb. The Hodge decomposition of η
η = dφη+ δβη+ αη, dφη∈ ◦
Bk, δβη∈ B∗k, αη ∈ ◦ Hk means for the trace that
tr η = d(tr φη) + tr(δβη) + tr αη= tr(δβη),
viz. the components dφη and αη of the extension η do not contribute to the tr η. Hence, we can construct η = δβη. Then, Lkη = δdδβη and it can be easily shown that hLkη, αi = 0 for all α ∈
◦ Hk.
On the other hand, given rN ∈ H1/2Λk−1(∂Ω), the extension result of Lemma 3.3.2 in18, guarantees the existence of ¯η ∈ H1ΛkΩ, such that
tr ¯η = 0 and tr(δ ¯η) = rN.
Take ¯u = u − η − ¯η. Then, Lku = f − L¯ kη − Lkη =: ¯¯ f , tr ¯u = 0, tr(δ ¯u) = 0 and using the condition (2.35), we can show that ¯f ⊥ H◦k. Hence, ¯u is the solution of the homogeneous boundary value problem (2.26)-(2.27) with the right hand side ¯f .
2.3.4. The Hodge Laplacian with inhomogeneous natural boundary conditions Consider the Hodge Laplacian operator Lk = dδ + δd : D(Lk) ⊂ L2Λk → L2Λk, with domain DN k =u ∈ HΛ k∩ H∗Λk| du ∈ H∗Λk+1, δu ∈ HΛk−1, tr(?u) = gb ∈ H−1/2Λn−k(∂Ω), tr(?du) = gN ∈ H−1/2Λn−k−1(∂Ω) o . (2.37) Our next step is to transform the inhomogeneous boundary value problem
(dδ + δd)u = f in Ω (2.38)
with the condition
hf, αi = − Z
∂Ω
tr α ∧ gN ∀α ∈ Hk, (2.40) and the side condition for uniqueness u ⊥ Hk, into the Hodge Laplacian homo-geneous boundary value problem (2.21). This can be considered as the dual of the problem with inhomogeneous essential boundary conditions, treated in Section 2.3.3. For completeness, we briefly summarize the steps of the construction.
For gb∈ H−1/2Λn−k(∂Ω) we can find τ ∈ H∗Λk(Ω) with tr(?τ ) = gb. Note here that since τ ∈ H∗Λk(Ω), then ?τ ∈ HΛn−k(Ω), so tr(?τ ) is well-defined. Moreover, using the Hodge decomposition
τ = dφτ+ δβτ+ ατ, dφτ∈ Bk, δβτ ∈ ◦
B∗k, ατ∈ Hk,
we have tr(?τ ) = tr(?dφτ), viz. the terms δβτand ατdo not contribute to the trace, hence we can take τ = dφτ. Then, Lkτ = dδdφτ and hLkτ, αi = 0 for all α ∈ Hk. Similarly, for gN ∈ H−1/2Λn−k−1(∂Ω), we can find ¯τ ∈ H∗Λk(Ω), with
?¯τ |∂Ω= 0, tr(δ ? ¯τ ) = gN.
Taking ¯u = u − τ − ¯τ , we obtain Lku = f − L¯ kτ − Lkτ =: ¯¯ f . Moreover, tr(?¯u) = 0, tr(?d¯u) = 0 and by using (2.40), we obtain ¯f ⊥ Hk.
Consequently, solving the inhomogeneous boundary value problem (2.38)-(2.40) for given f ∈ L2Λk is equivalent with solving the Hodge Laplacian problem with homogeneous boundary conditions (2.21), for given ¯f .
Note that, since the B and B∗problems are special cases of the Hodge Laplacian problem, they can be solved also for inhomogeneous boundary conditions.
3. The Euler equations in density weighted vorticity and dilatation formulation
In this section we will derive, via the Hodge decomposition, a Hamiltonian formu-lation of the compressible Euler equations using a density weighted vorticity and dilatation as primary variables. This will provide an extension of the well known vorticity-streamfunction formulation of the incompressible Euler equations to com-pressible flows, see e.g.12,13. Special attention will be given to the proper boundary conditions for the potential φ and the vector stream function β.
The analysis of the Hamiltonian formulation and Stokes-Dirac structure of the compressible Euler equations is most easily performed using techniques from dif-ferential geometry. For this purpose we first reformulate (1.1)-(1.3) in terms of differential forms. We identify the mass density ρ and the entropy s with a 3-form on Ω, that is, with an element of Λ3(Ω), the pressure p ∈ Λ0(Ω), and the velocity field u with a 1-form on Ω, viz., with an element of Λ1(Ω). Let u]be the vector field corresponding to the 1-form u (using index raising, or sharp mapping), iu] denotes
For an arbitrary vector field X and α ∈ Λk(Ω), the following relation between the interior product and Hodge star operator is valid, (see e.g.11),
iXα = ?(X[∧ ?α), (3.1)
where X[ is the 1-form related to X by the flat mapping. Following the identifica-tions of the variables suggested in17for the isentropic fluid using differential forms, and completing them with the entropy balance equation, the Euler equations of gas dynamics can then be formulated in differential forms as
ρt= −d(iu]ρ), (3.2) ut= −d 1 2u ], u] v − iu]du − 1 ?ρd p (3.3) st= −u ∧ ?d(?s) = u ∧ δs, (3.4)
where h·, ·iv is the inner product of two vectors.
Using the L2-de Rham complex described in Section 2.2, let’s start with the Hodge decomposition of the differential 1-form √ρ ∧ u ∈ L˜ 2Λ1(Ω), denoted by ζ,
ζ =pρ ∧ u = dφ + δβ + α,˜ (3.5)
where ˜ρ = ?ρ. The choice of ζ will be motivated in the next section.
Definition 3.1. Using the Hodge decomposition (3.5), define the density weighted vorticity as ω = dζ and the density weighted dilatation as θ = −δζ.
There are two Hodge decompositions, (2.11) and (2.16), hence two sets of bound-ary conditions on the Hodge components
(a) dφ ∈ B1, δβ ∈ ◦
B∗1, α ∈ H1, (b) dφ ∈B◦1, δβ ∈ B∗1, α ∈H◦1.
Remark 3.1. If the classical vorticity is ξ = du then, the density weighted vorticity is
ω = dζ = d(pρ ∧ u) = (d˜ √?ρ) ∧ u +√?ρ ∧ ξ. (3.6) When the flow is incompressible, then ω = √?ρ ∧ ξ, i.e., the density weighted vorticity. Using the equation for the velocity (3.3), the vorticity evolution equation for constant density is
ξt= −d (iu]du) = −d (iu]ξ) . (3.7)
On the other hand, the evolution equation for density weighted vorticity for incom-pressible flows is
ωt= √
?ρ ∧ ξt= −d (iu]ω) . (3.8)
The density weighted dilatation θ for an incompressible flow is
θ = −δ (√?ρ ∧ u) = −√?ρ ∧ δu. (3.9) The incompressibility constraint in differential forms is δu = 0, which implies θ = 0. Lemma 3.1. The potential function φ and vector stream function β in the Hodge decomposition (3.5) solve the following boundary value problems
B-problem ( dδβ = ω, dβ = 0 in Ω tr(?β) = 0 on ∂Ω, B ∗-problem ( δdφ = −θ in Ω tr(?dφ) = 0 on ∂Ω, (3.10) when ζ ∈ HΛ1(Ω) ∩ ◦ H∗Λ1(Ω) ∩ H1⊥ and B-problem ( dδβ = ω, dβ = 0 in Ω tr(δβ) = 0 on ∂Ω, B ∗-problem ( δdφ = −θ in Ω tr(φ) = 0 on ∂Ω, (3.11) when ζ ∈ ◦ H Λ1(Ω) ∩ H∗Λ1(Ω) ∩H◦1⊥.
Proof. The proof of this lemma is constructive and we consider two cases. Case 1. The first approach is to choose ζ ∈ HΛ1(Ω) ∩H◦∗Λ1(Ω) ∩ H1⊥. Then, the Hodge decomposition (a) of ζ is used to define two new variables.
Since ζ ∈ D(d) = HΛ1(Ω), we can define the density weighted vorticity ω ∈ B2⊂ L2Λ2(Ω) as
ω = dζ = d(pρ ∧ u) = d(dφ + δβ) = dδβ,˜ (3.12) where φ ∈ HΛ0(Ω), β ∈ H∗Λ2(Ω) with tr(?β) = 0 and δβ ∈ D(d) = HΛ1(Ω). Moreover, since ω ∈ B2, it follows that β ∈ B2 hence, dβ = 0. Observe here, that (3.12) is the B problem (2.22)-(2.23) with homogeneous natural boundary conditions.
Consider now ζ ∈ D(δ) = H◦∗Λ1(Ω). Define the density weighted dilatation θ ∈ B∗0⊂ L2Λ0(Ω) as
θ = −δζ = −δ(pρ ∧ u) = −δdφ − δδβ = −δdφ,˜ (3.13) where φ ∈ HΛ0(Ω), β ∈H◦∗Λ2(Ω) and dφ ∈ H∗Λ1(Ω) with 0 = tr(?ζ) = tr(?dφ). Observe that (3.13) with this boundary condition is the B∗ problem (2.24)-(2.25) for φ, where δφ = 0 is satisfied since HΛ−1 is understood to be zero and tr(?φ) = 0 since ?φ is a 3-form.
Case 2. Let us choose now ζ ∈ ◦
H Λ1(Ω) ∩ H∗Λ1(Ω) ∩H◦1⊥. Then, we use the Hodge decomposition (b) of ζ to define two new variables.
Since ζ ∈ ◦
H Λ1(Ω), tr ζ = 0 and using the decomposition in (b), we obtain the weakly imposed essential boundary condition 0 = tr ζ = tr(δβ). Define the density weighted vorticity ω ∈B◦2as
where β ∈ H∗Λ2(Ω). This is the B problem (2.29)-(2.30) for β with homogeneous essential boundary conditions. Similarly, since ζ ∈ H∗Λ1(Ω), we can define the density weighted dilatation θ ∈ B∗0, as
θ = −δdφ, (3.15)
where φ ∈ ◦
H Λ0(Ω), dφ ∈ H∗Λ1(Ω). The Hodge decomposition in (b) implies the strongly imposed boundary condition tr φ = 0. This is again the B∗problem (2.31)-(2.32) for φ, with homogeneous essential boundary conditions.
Remark 3.2. When the flow is incompressible, then 0 = θ = −δζ = −δdφ. This implies that
0 = − hδdφ, φi = − hdφ, dφi + Z
∂Ω
tr φ ∧ tr(?dφ) = − hdφ, dφi .
The boundary integral is zero in either of the two Hodge decompositions. Then, dφ = 0 hence, the Hodge decomposition is ζ = δβ + α and the B∗-problems in Lemma 3.1 cancel. The B-problems remain unchanged.
Remark 3.3. Note, in Lemma 3.1, we can consider inhomogeneous boundary conditions for all problems. More precisely, for Case 1, the B problem for β has a unique solution β ∈ B2, with the inhomogeneous boundary condition tr(?β) = gb ∈ H−1/2Λ1(∂Ω), by transforming it first to a homogeneous problem with modified right hand side δdβ = ¯ω, as we discussed in Section 2.3.4. The B∗ problem for φ has a unique solution with the inhomogeneous boundary condition tr(?dφ) = gN ∈ H−1/2Λ2(∂Ω). Hence, solving the B∗ problem (3.13) with inho-mogeneous boundary conditions, is equivalent with solving the B∗ homogeneous problem with modified right hand side δdφ = −¯θ, with ¯θ = θ + ¯τ , as seen in Section 2.3.4.
In Case 2, the inhomogeneous essential boundary conditions for φ and β are tr φ = rb ∈ H1/2Λ0(∂Ω) and tr(δβ) = rN ∈ H1/2Λ1(∂Ω), respectively. We can transform the equations into a homogeneous problem as in Section 2.3.3.
Corollary 3.1. The non-isentropic compressible Euler equations can be formulated in the variables ρ, ω, θ and s as
ρt= −d( p ˜ ρ ∧ ?ζ), (3.16) ωt= d ζt, (3.17) θt= −δζt, (3.18) st= 1 √ ˜ ρ∧ ζ ∧ δs, (3.19) with ζ given by (3.5).
Proof. The statement of this Corollary can easily be verified by introducing the Hodge decomposition (3.5) into the Euler equations (3.2-3.4).
Summarizing, we use the Hodge decomposition (3.5) to define the density weighted vorticity ω and dilatation θ. In order to have them well defined, we choose the proper spaces for ζ. The potential function φ and vector stream function β in the Hodge components are the solutions of the B∗ and B problems, respectively, with natural or essential boundary conditions.
4. Functional derivatives of the Hamiltonian in density weighted vorticity-dilatation formulation
In this section we transform the Hamiltonian functional for the non-isentropic com-pressible Euler equations, into the new set of variables ρ, θ, ω, s, and calculate the variational derivatives with respect to these new variables.
In order to simplify the discussion, we assume from here on that the domain Ω is simply connected. Since the dimension of H1 is equal to the first Betti number, i.e., the number of handles, of the domain, it follows that H1 = 0 if the domain is simply connected, see3. Then α = 0 in the Hodge decomposition (3.5).
Let us recall from van der Schaft and Maschke17the definition of the variational derivatives of the Hamiltonian functional when it depends on, for example, two energy variables. Consider a Hamiltonian density, i.e. energy per volume element,
H : Λp(Ω) × Λq(Ω) → Λn(Ω), (4.1) where Ω is an n-dimensional manifold, resulting in the total energy
H[αp, αq] = Z
Ω
H(αp, αq) ∈ R, (4.2)
where square brackets are used to indicate that H is a functional of the enclosed functions. Let αp, ∂αp∈ Λp(Ω), and αq, ∂αq ∈ Λq(Ω). Then under weak smoothness assumptions on H, H[αp+ ∂αp, αq+ ∂αq] = Z Ω H(αp+ ∂αp, αq+ ∂αq) = Z Ω H(αp, αq) + Z Ω (δpH ∧ ∂αp+ δqH ∧ ∂αq) + higher order terms in ∂αp, ∂αq, (4.3) for certain uniquely defined differential forms δpH ∈ (Λp(Ω))∗and δqH ∈ (Λq(Ω))∗, which can be regarded as the variational derivatives of H with respect to αp and αq, respectively. The dual linear space (Λp(Ω))∗ can be naturally identified with Λn−p(Ω), and similarly the dual space (Λq(Ω))∗ with Λn−q(Ω).
For the non-isentropic compressible Euler equations, the energy density is given as the sum of the kinetic energy and internal energy densities. The Hamiltonian functional for the compressible Euler equations in differential forms is (see17)
H[ρ, u, s] = Z Ω 1 2u ], u] vρ + U ( ˜ρ, ˜s)ρ , (4.4)
where ˜s = ?s. The Hamiltonian functional can further be written as H[ρ, u, s] = Z Ω 1 2 D (pρ ∧ u)˜ ], (pρ ∧ u)˜ ]E v vΩ+ U ( ˜ρ, ˜s)ρ = Z Ω 1 2 p ˜ ρ ∧ u,pρ ∧ u v˜ Ω+ U ( ˜ρ, ˜s)ρ . (4.5)
The Hamiltonian written in form (4.5) motivates the choice of the variable ζ, de-fined in (3.5), and its Hodge decomposition. In the next lemma we compute the Hamiltonian functional and its variational derivatives with respect to the Hodge decomposition of ζ.
Remark 4.1. We have seen that the inhomogeneous B∗ problem for φ and the inhomogeneous B problem for β can be transformed into homogeneous boundary value problems with modified right hand side. Therefore, from here on we just use the standard de Rham theory for the bar variables ¯θ and ¯ω, with the corresponding homogeneous boundary conditions for the φ and β variables.
Lemma 4.1. The Hamiltonian density, when the variables ρ, φ, β, s are introduced, is a mapping
H : HΛ3× D0× D2× HΛ3→ L2Λ3, (ρ, φ, β, s) 7→ H(ρ, φ, β, s),
where D0and D2 are the domains of the Hodge Laplacian for 0-forms and 2-forms, respectively, with either essential or natural inhomogeneous boundary conditions, as defined in (2.36) and (2.37). This results in the total energy
H[ρ, φ, β, s] = Z Ω 1 2(δdφ ∧ ?φ + dδβ ∧ ?β) + U ( ˜ρ, ˜s)ρ . (4.6)
The variational derivatives of the Hamiltonian are δH δρ = ∂ ∂ ˜ρ( ˜ρU ( ˜ρ, ˜s)), δH δφ = − ? ¯θ, δH δβ = ?¯ω, δH δs = ∂U ( ˜ρ, ˜s) ∂ ˜s ρ.˜ (4.7) Proof. Introducing the Hodge decomposition (3.5) into the Hamiltonian in (4.5) and using the inner product (2.1), we obtain the following Hamiltonian when the variables ρ, φ, β, s are introduced
H[ρ, φ, β, s] = 1 2 Z Ω dφ + δβ, dφ + δβ vΩ+ Z Ω U ( ˜ρ, ˜s)ρ = 1 2hdφ + δβ, dφ + δβi + Z Ω U ( ˜ρ, ˜s)ρ. (4.8) Using the definition of the density weighted vorticity and dilatation, after partial integration, the inner product in the Hamiltonian in (4.8) reduces to
hdφ + δβ, dφ + δβi = hdφ, dφi + hdφ, δβi + hδβ, δβi + hδβ, dφi = hφ, δdφi + hβ, dδβi =φ, −¯θ + hβ, ¯ωi ,
where hdφ, δβi = 0 and hδβ, dφi = 0 since (3.5) is an orthogonal decomposition. Note that this is valid for both types of boundary conditions, since in either case the boundary integrals are zero. Introducing this inner product into (4.8) we obtain (4.6).
Consider (4.8) in the form H[ρ, φ, β, s] =1
2(hdφ, dφi + hδβ, δβi) + Z
Ω
U ( ˜ρ, ˜s)ρ, (4.9) where the inner products are in the space L2Λ1(Ω). Let us calculate δφH first. For φ, ∂φ ∈ D(L0) and β ∈ D(L2), with either essential or natural homogeneous boundary conditions on φ and ∂φ, we have
H[ρ, φ + ∂φ, β, s] = Z Ω H(ρ, φ, β, s) + hdφ, d(∂φ)i + {h. o. t. in ∂φ} = Z Ω H(ρ, φ, β, s) + hδdφ, ∂φi + {h. o. t. in ∂φ} Therefore, δH δφ = ?δdφ = −d(?dφ) = − ? ¯θ. (4.10) Similarly, let us calculate the variational derivative δβH. For β, ∂β ∈ D(L2), we have for either boundary conditions,
H[ρ, φ, β + ∂β, s] = Z
Ω
H(ρ, φ, β, s) + h∂β, dδβi + {h. o. t. in ∂β}. (4.11) Hence, we obtain that
δH
δβ = ?dδβ = ?¯ω, (4.12)
which completes the proof of this lemma. The variational derivatives of the Hamil-tonian with respect to the variables ρ and s, given in (4.7), can easily be calculated, see17
Remark 4.2. We have defined the problem in the bar variables to account for inhomogeneous boundary conditions, see Section 2.3. From here on we drop the bar to make the notation simpler.
Our aim is now to formulate the Hamiltonian as a functional of ρ, θ, ω, s and calculate the variational derivatives with respect to these new variables.
Lemma 4.2. The Hamiltonian density in the variables ρ, θ, ω, s is a mapping H : HΛ3× B∗
0× B
2× HΛ3→ L2Λ3,
which results in the total energy H[ρ, θ, ω, s] = Z Ω 1 2(θ ∧ ?K0θ + ω ∧ ?K2ω) + U ( ˜ρ, ˜s)ρ , (4.14) where Kk is the solution operator of the Hodge Laplacian operator Lk, k = 0, 2. The variational derivatives of the Hamiltonian functional are:
δH δρ = ∂ ∂ ˜ρ( ˜ρU ( ˜ρ, ˜s)), δH δω = ?K2ω = ?β, (4.15) δH δθ = ?K0θ = − ? φ, δH δs = ∂U ( ˜ρ, ˜s) ∂ ˜s ρ.˜ (4.16)
Proof. Using Lemma 4.1, and that θ and ω are in the domain of the solution operators K0 and K2 of the Hodge Laplacian problems for φ and β, respectively, the Hamiltonian can be written as
H[ρ, θ, ω, s] = Z Ω 1 2(θ ∧ ?K0θ + ω ∧ ?K2ω) + U ( ˜ρ, ˜s)ρ . (4.17) Let θ, ∂θ ∈ B∗0 and ω, ∂ω ∈ B2. The variational derivatives of the Hamiltonian with respect to θ and ω can be obtained from
H[ρ, θ + ∂θ, ω + ∂ω, s] = 1 2 Z Ω (θ + ∂θ) ∧ ?K0(θ + ∂θ) + Z Ω U ( ˜ρ, ˜s)ρ +1 2 Z Ω (ω + ∂ω) ∧ ?K2(ω + ∂ω) = Z Ω H(ρ, θ, ω, s) +1 2 Z Ω (θ ∧ ?K0(∂θ) + ∂θ ∧ ?K0θ) +1 2 Z Ω (ω ∧ ?K2(∂ω) + ∂ω ∧ ?K2ω) + { h. o. t. in ∂θ, ∂ω}. (4.18)
Here ∂θ and ∂ω denote the variation of θ and ω, respectively, to avoid confusion with the coderivative operator δ. In order to further investigate the last two integrals in (4.18), we use Lemma 4.1, where the variational derivatives of the Hamiltonian with respect to φ and β are given, then apply the variational chain rule to obtain δθH and δωH.
To obtain the variational derivative of the Hamiltonian with respect to θ, apply the variational chain rule as follows
Z Ω δH δφ ∧ ∂φ = Z Ω δH δθ ∧ ∂θ = − Z Ω δH δθ ∧ δd(∂φ) = − ?δH δθ, δd(∂φ) = − Z Ω dδ(δH δθ ) ∧ ∂φ + Z ∂Ω tr(∂φ) ∧ tr(δδH δθ) + Z ∂Ω tr(?d(∂φ)) ∧ tr(?δH δθ), (4.19)
where ∂φ, ∂θ = −δd(∂φ) denote the variations of φ and θ, to avoid confusion with the coderivative operator δ. Analogously, to obtain the variational derivative of the Hamiltonian with respect to ω, consider the variational chain rule
Z Ω δH δβ ∧ ∂β = Z Ω δH δω ∧ ∂ω = Z Ω δH δω ∧ dδ(∂β) = ?δH δω, dδ(∂β) = Z Ω δd(δH δω) ∧ ∂β + Z ∂Ω tr(δ(∂β)) ∧ trδH δω − Z ∂Ω tr(?∂β) ∧ tr(?dδH δω). (4.20) We now have two choices.
First, when Case 1 applies in Lemma 3.1, then tr(?d(∂φ)) = 0, hence the vari-ational equation (4.19) becomes
Z Ω δH δφ ∧ ∂φ = − Z Ω dδ(δH δθ) ∧ ∂φ + Z ∂Ω tr(∂φ) ∧ tr(δδH δθ), (4.21) ∀ ∂φ ∈ HΛ0(Ω), with tr(?d(∂φ)) = 0. Choose ∂φ such that the boundary integral in (4.21) is zero. We thus obtain that δH
δθ solves the differential equation dδ(δH
δθ) = − δH
δφ = ?θ in Ω. (4.22)
Consider now the variation ∂φ ∈ D(L0) arbitrary. Inserting (4.22) into the varia-tional equation (4.21), we obtain that
tr(δδH
δθ) = 0, (4.23)
which together with (4.22) is precisely the B problem with essential boundary conditions (2.29)-(2.30) for δHδθ, with weakly imposed boundary condition (4.23). On the other hand, combining (4.22) with (4.10) leads to
δH
δθ = − ? φ + h, (4.24)
with h ∈ Z∗3, the null space of δ. The B problem for δHδθ has however, a unique solution δHδθ ∈B◦3, hence the side condition δHδθ ⊥ Z∗
3 is satisfied. Consequently, δH
δθ = − ? φ, (4.25)
where φ is the unique solution of the B∗ problem (2.24)-(2.25).
We still need to calculate δωH, when Case 1 applies. Since tr(?∂β) = 0, the last integral in (4.20) cancels. Using the same arguments as before, we obtain the B∗ problem with essential boundary condition
δd(δH δω) = δH δβ in Ω, tr( δH δω) = 0 on ∂Ω. (4.26)
Combined with (4.12), we obtain the following equation δd(δH
which leads to δH
δω = ?β + h, with h ∈ ◦
Z1, the null space of d. On the other hand, the B∗ problem (4.26) has a unique solution δH
δω ∈ B ∗ 1= ◦ Z1⊥. Therefore, δH δω = ?β, (4.28)
where β is the unique solution of the B problem (2.22)-(2.23).
When Case 2 applies, the first boundary integral in both variational equations (4.19) and (4.20) will be zero. In a completely analogous way as in Case 1, we obtain the following B problem for δHδθ when natural boundary conditions hold
dδ(δH
δθ) = ?θ in Ω, tr(? δH
δθ) = 0 on ∂Ω, (4.29)
and the B∗ problem with natural boundary condition for δH δω
δd(δH
δω) = ?ω in Ω, tr(?d δH
δω) = 0 on ∂Ω. (4.30)
Analyzing the solution of these problems leads to the same variational derivatives (4.25) and (4.28).
The variational derivatives of the Hamiltonian with respect to the variables ρ and s, given in (4.15) and (4.16), respectively, can easily be calculated, see 17.
Summarizing, when φ and β solve a B∗ and B problem, respectively, with (in)homogeneous natural or essential boundary conditions, the variational deriva-tives δH
δθ and δH
δω will solve a dual problem, viz. a B and B
∗ problem, respectively, with the corresponding (dual) boundary conditions.
5. Poisson bracket
The nonlinear system (3.2)-(3.4) has a Hamiltonian formulation with the Poisson bracket of Morrison and Green 13,12 with the Hamiltonian given by (4.4) in the ρ, u, s variables, (see also17). The Poisson bracket has the form
{F , G} = − Z Ω δF δρ ∧ d δG δu − δG δρ ∧ d δF δu | {z } T1 + ? iX ? ?δG δu∧ ? δF δu | {z } T2 + 1 ˜ ρ∧ d˜s ∧ δF δs ∧ δG δu− δG δs ∧ δF δu , | {z } T3 (5.1)
with the vector field X = (?duρ˜ )]. The aim of this section is to transform the Poisson bracket into the new set of variables ρ, θ, ω, s and to properly account for
the boundary conditions. We derive the bracket in a well-chosen functional space using the functional chain rule.
Lemma 5.1. Consider the Hodge decomposition of ζ in (3.5) and let F [ρ, θ, ω, s] and G[ρ, θ, ω, s] be arbitrary functionals. Assume that
ζ ∈ ◦ H Λ1(Ω) ∩ H∗Λ1(Ω) ∩H◦1⊥, and tr(?δF δθ) = 0, tr(? δG δθ) = 0 (5.2) or ζ ∈ HΛ1(Ω) ∩ ◦ H∗Λ1(Ω) ∩ H1⊥, and tr(δF δω) = 0, tr( δG δω) = 0. (5.3) Then, the bracket (5.1) in terms of the variables ρ, θ, ω, s has the form
{F , G} = − Z Ω δF δρ ∧ d p ˜ ρ ∧ α(G)−δG δρ ∧ d p ˜ ρ ∧ α(F ) + ? ζ 2 ˜ρ∧ α(F ) ∧ dpρ ∧ α(G)˜ − ? ζ 2 ˜ρ∧ α(G) ∧ dpρ ∧ α(F )˜ + √d˜s ˜ ρ∧ δF δs ∧ α(G) − δG δs ∧ α(F ) , (5.4)
where α(·) is the following operator on functionals α(·) = dδ ·
δω+ δ δ ·
δθ. (5.5)
Proof. The transformation of the bracket requires the use of the chain rule for functional derivatives. First, we would like to know how the value of ζ[ρ, u], given by (3.5), changes as ρ and u are slightly perturbed, say ρ → ρ+∂ρ and u → u+∂u. The first variation ∂ζ of ζ induced by ∂ρ is given by (see e.g.12)
∂ζ[ρ; ∂ρ, u] = d dζ[ρ + ∂ρ, u] =0= u 2√ρ˜∧ ?∂ρ. (5.6) Similarly, the variation ∂ζ of ζ induced by ∂u is given by
∂ζ[ρ, u; ∂u] = d dζ[ρ, u + ∂u] =0= p ˜ ρ ∧ ∂u. (5.7)
Hence, the total variation ∂ζ of ζ is ∂ζ[ρ, u; ∂ρ, ∂u] = u
2√ρ˜∧ ?∂ρ + p
˜
ρ ∧ ∂u. (5.8)
We can define a functional of ρ, θ, ω, s by introducing the Hodge decomposition (3.5) into F [ρ, u, s] and obtain ¯F [ρ, θ, ω, s] = F [ρ, u, s]. This means that the following variational equation holds
?δF δρ, ∂ρ + ?δF δu, ∂u = ?δ ¯F δρ, ∂ρ + ?δ ¯F δθ, ∂θ + ?δ ¯F δω, ∂ω , (5.9)
where ∂θ = −δ(∂ζ) and ∂ω = d(∂ζ). By partial integration, we obtain ?δ ¯F δθ, ∂θ = − ?δ ¯F δθ, δ(∂ζ) = − d ?δ ¯F δθ, ∂ζ + Z ∂Ω tr(?δ ¯F δθ) ∧ tr(?∂ζ) = Z Ω δδ ¯F δθ ∧ ∂ζ = Z Ω δδ ¯F δθ ∧ u 2√ρ˜∧ ?∂ρ + δ δ ¯F δθ ∧ p ˜ ρ ∧ ∂u , where the boundary integral cancels, in case of decomposition (a) because tr(?∂ζ) = 0, and in case of decomposition (b) because of the assumption tr(?δFδθ) = 0. Simi-larly, we obtain by partial integration that
?δ ¯F δω, ∂ω = ?δ ¯F δω, d(∂ζ) = δ ?δ ¯F δω, ∂ζ + Z ∂Ω tr(∂ζ) ∧ tr(δ ¯F δω) = Z Ω dδ ¯F δω ∧ u 2√ρ˜∧ ?∂ρ + d δ ¯F δω ∧ p ˜ ρ ∧ ∂u .
Here the boundary integral cancels, in case of decomposition (a) because of the assumption tr(δFδω) = 0, and in case of decomposition (b) because tr(∂ζ) = 0. Since the variational equation (5.9) holds for all ∂ρ, ∂u, we obtain the relations
δF δρ ρ,u,s = δ ¯F δρ ρ,ω,θ,s + ? u 2√ρ˜∧ dδ ¯F δω + δ δ ¯F δθ (5.10) δF δu = p ˜ ρ ∧ dδ ¯F δω + δ δ ¯F δθ . (5.11)
From here on we will drop the overbar on F . We can write the functional chain rules in (5.10) and (5.11) as δF δρ ρ,u,s = δF δρ ρ,ω,θ,s + ? ζ 2 ˜ρ∧ α(F ) , (5.12) δF δu = p ˜ ρ ∧ α(F ). (5.13)
Introducing the relations above into the bracket (5.1) we obtain (5.4), which com-pletes the proof of the lemma.
Remark 5.1. It is straightforward to verify that the bracket in (5.4) is skew-symmetric. The Jacobi identity is also satisfied, see 13.
In the next technical lemma we give the integrated-by-parts form of the bracket (5.4). This is only done to make it easier to verify the statement of Theorem 5.1. The proof is given in Appendix A.
bracket in (5.4) can be transformed into {F , G} = − Z Ω δF δρ ∧ d p ˜ ρ ∧ α(G)+δF δs ∧ 1 √ ˜ ρ∧ d˜s ∧ α(G) +δF δω ∧ dγ(gradG) − δF δθ ∧ δγ(gradG) + Z ∂Ω trpρ ∧ α(F )˜ ∧ tr δG δρ + ? α(G) ∧ ζ 2 ˜ρ , (5.14) where gradG = (δρG, δωG, δθG, δsG). (5.15) Furthermore, γ(gradG) =ζ 2 ˜ρ∧ ?d( p ˜ ρ ∧ α(G)) +pρ ∧ d˜ δG δρ + ?(α(G) ∧ ζ 2 ˜ρ) + ? ?d ζ √ ˜ ρ ∧ ?α(G) −√d˜s ˜ ρ∧ δG δs. (5.16)
Remark 5.2. The skew-symmetry of the bracket (5.14) can be directly checked by using partial integration.
Theorem 5.1. The equations of motion for the density ρ, vorticity ω, dilatation θ and entropy s, given by (3.16), (3.17), (3.18) and (3.19) respectively, are obtained from the bracket (5.14) as
∂ρ ∂t = {ρ, H}, ∂ω ∂t = {ω, H} − Z ∂Ω tr(pρ ∧ α(ω)) ∧ tr˜ δH δρ + ? ζ 2 ˜ρ∧ α(H) , ∂s ∂t = {s, H}, ∂θ ∂t = {θ, H} − Z ∂Ω tr(pρ ∧ α(θ)) ∧ tr˜ δH δρ + ? ζ 2 ˜ρ∧ α(H) , with H the Hamiltonian given by (4.14).
Proof. The proof of this theorem is very straightforward if we observe that α(H) = ?ζ and γ(grad H) = −ζt. (5.17)
6. Stokes-Dirac structures
The treatment of infinite dimensional Hamiltonian systems in the literature seems mostly focused on systems with an infinite spatial domain, where the variables go to zero for the spatial variables tending to infinity12, or on systems with boundary conditions such that the energy exchange through the boundary is zero. It is, how-ever essential from an application point of view to describe a system with varying boundary conditions, including energy exchange through the boundary. In Remark
3.3 we already indicated how to treat inhomogeneous boundary conditions in the Hodge decomposition, which provide an essential ingredient for the Hamiltonian formulation. An alternative approach is to follow van der Schaft and Maschke 16, where a framework to overcome the difficulty of incorporating non-zero energy flow through the boundary in the Hamiltonian framework for distributed-parameter sys-tems is presented. This is done by using the notion of a Stokes-Dirac structure16,17. We will demonstrate that both approaches lead to the same formulation. A general definition of a Stokes-Dirac structure is given as follows.
Definition 6.1. Let F be a linear space (finite or infinite dimensional). There exists on F × E the canonically defined symmetric bilinear form
(f1, e1), (f2, e2)
D:=e1, ?f2 + e2, ?f1 = Z
Ω
(e1∧ f2+ e2∧ f1), with fi∈ F , ei ∈ E, i = 1, 2, and h·, ·i denoting the duality pairing between F and its dual space E . A Stokes-Dirac structure on F is a linear subspace D ⊂ F × E , such that
D = D⊥,
where ⊥ denotes the orthogonal complement with respect to the bilinear form , D.
6.1. Stokes-Dirac structure for the non-isentropic compressible Euler equations
The Stokes-Dirac structure for distributed-parameter systems used in17has a spe-cific form by being defined on spaces of differential forms on the spatial domain of the system and its boundary. The construction of the Stokes-Dirac structure empha-sizes the geometrical content of the physical variables involved, by identifying them as appropriate differential forms. In17the description is given for the compressible Euler equations for an ideal isentropic fluid.
In this section we first extend the Stokes-Dirac structure given in 17 to the non-isentropic Euler equations (1.1)-(1.3). In Section 6.3 we derive the Stokes-Dirac structure for the non-isentropic Euler equations in the ρ, ω, θ, s variables. The linear spaces on which the Stokes-Dirac structure for the ρ, u, s variables will be defined are:
F : = Λ3(Ω) × Λ1(Ω) × Λ3(Ω) × Λ0(∂Ω), (6.1) E : = Λ0(Ω) × Λ2(Ω) × Λ0(Ω) × Λ2(∂Ω). (6.2) The following theorem is an extension of Theorem 2.1 in17 for the non-isentropic Euler equations. In the proof, we closely follow the proof of Theorem 2.1 in 17. Theorem 6.1. (Stokes-Dirac structure) Let Ω ⊂ R3 be a three dimensional mani-fold with boundary ∂Ω. Consider F and E as given by (6.1) and (6.2) respectively,
together with the bilinear form (f1, e1), (f2, e2) D= Z Ω (e1ρ∧ f2 ρ + e 2 ρ∧ f 1 ρ+ e 1 u∧ f 2 u+ e 2 u∧ f 1 u + e1s∧ f2 s + e 2 s∧ f 1 s) + Z ∂Ω (e1b∧ fb2+ e 2 b∧ f 1 b) , (6.3) where fi= (fρi, fui, fsi, fbi) ∈ F , ei= (eiρ, eiu, eis, eib) ∈ E , i = 1, 2. Then D ⊂ F × E defined as D ={((fρ, fu, fs, fb), (eρ, eu, es, eb)) ∈ F × E | fρ= deu, fs= 1 ˜ ρd˜s ∧ eu, fu= deρ− 1 ˜ ρd˜s ∧ es+ 1 ˜
ρ? ((?du) ∧ (?eu)), fb= tr(eρ), eb= −tr(eu), (6.4) is a Stokes-Dirac structure with respect to the bilinear form , Ddefined in (6.3).
Proof. The proof of this theorem consists of two steps.
Step 1. First we show that D ⊂ D⊥. Let (f1, e1) ∈ D be fixed, and consider any (f2, e2) ∈ D. Substituting the definition of D into (6.3), we obtain that
I : = (f1, e1), (f2, e2) D = Z Ω e1 ρ∧ de 2 u+ e 2 ρ∧ de 1 u+ e 1 u∧ (de 2 ρ+ T 2 + S2) + e2u∧ (de 1 ρ+ T 1 + S1) +e1s∧ 1 ˜ ρd˜s ∧ e 2 u+ e 2 s∧ 1 ˜ ρd˜s ∧ e 1 u + Z ∂Ω e1b∧ tr(e2 ρ) + e 2 b∧ tr(e 1 ρ) , (6.5)
where Ti = ρ1˜? ((?du) ∧ (?eui)) and Si = −1ρ˜d˜s ∧ e i
s are 1-forms, for i = 1, 2. Regrouping the terms, we can write the integral (6.5) as
I = Z Ω (e1ρ∧ de2 u+ e 2 u∧ de 1 ρ) | {z } I1 + (e2ρ∧ de1 u+ e 1 u∧ de 2 ρ) | {z } I2 + (e1u∧ T2+ e2 u∧ T 1) | {z } I3 + Z Ω e1u∧ S 2 + e2u∧ S 1 + e1s∧ 1 ˜ ρd˜s ∧ e 2 u+ e 2 s∧ 1 ˜ ρd˜s ∧ e 1 u | {z } I4 + Z ∂Ω e1b∧ tr(e2ρ) + e 2 b∧ tr(e 1 ρ) .
Using the properties of the wedge product and the Leibniz rule for exterior differ-entiation, we obtain I1= Z Ω (de2u∧ e1 ρ+ e 2 u∧ de 1 ρ) = Z Ω d(e2u∧ e1 ρ) = Z ∂Ω tr(e2u∧ e1 ρ), I2= Z Ω (de1u∧ e2 ρ+ e 1 u∧ de 2 ρ) = Z Ω d(e1u∧ e2 ρ) = Z ∂Ω tr(e1u∧ e2 ρ). Observe that F (e1u, e2u) := Z Ω e1u∧ T2= Z Ω e1u∧1 ˜ ρ∧ ?(?du ∧ ?e 2 u) (6.6) is skew-symmetric in e1
u, e2u∈ Λ2(Ω), that is, F (e1u, e2u) = −F (e2u, e1u). This implies that I3= 0. Furthermore, introducing Si into I4 we obtain that I4= 0. Therefore,
I = Z
∂Ω
tr(e2u) ∧ tr(e1ρ) + tr(eu1) ∧ tr(e2ρ) + e1b∧ tr(e2 ρ) + e
2 b∧ tr(e
1 ρ) = 0, where the last equality is true since in D we have eib= − tr(eiu), i = 1, 2. We proved that the bilinear form (6.3) is zero for all (f2, e2) ∈ D. Hence, (f2, e2) ∈ D⊥.
Step 2. Next we show that D⊥⊂ D. Let (f1, e1) ∈ D⊥. Then,
J := (f1, e1), (f2, e2) D= 0, ∀(f2, e2) ∈ D. (6.7) Since (f2, e2) ∈ D, (6.7) is equivalent to J = Z Ω e1ρ∧ de 2 u+ e 2 ρ∧ f 1 ρ+ e 1 u∧ de 2 ρ+ e 1 u∧ T 2 + e1u∧ S 2 + e2u∧ f 1 u +e1s∧1 ˜ ρd˜s ∧ e 2 u+ e 2 s∧ f 1 s + Z ∂Ω (e1b∧ tr(e2 ρ) − tr(e 2 u) ∧ f 1 b) = 0, ∀e 2 ρ∈ Λ 0(Ω), e2 u∈ Λ 2(Ω), e2 s∈ Λ 0(Ω). (6.8) Take e2ρ ∈ Λ0(Ω), e2u ∈ Λ2(Ω), such that tr(e2ρ) = 0 and tr(e2u) = 0. Then, the boundary integral in J vanishes. Using the Leibniz rule for the underlined terms, we obtain J = Z Ω d(e1 ρ∧ e 2 u) − de 1 ρ∧ e 2 u+ f 1 ρ ∧ e 2 ρ+ d(e 1 u∧ e 2 ρ) − de 1 u∧ e 2 ρ+ f 1 u∧ e 2 u + e1u∧ T2+ e1 u∧ S 2+ e1 s∧ 1 ˜ ρd˜s ∧ e 2 u+ e 2 s∧ f 1 s = 0,
for all e2ρ ∈ Λ0(Ω), e2u ∈ Λ2(Ω), e2s ∈ Λ0(Ω) with tr(e2ρ) = 0 and tr(e2u) = 0. Using Stokes’ theorem, the assumptions on e2
ρ, e2u, the skew-symmetry of F (e1u, e2u) in (6.6) and e1u∧ S2= e1 u∧ (− 1 ˜ ρd˜s ∧ e 2 s) = − 1 ˜ ρd˜s ∧ e 1 u∧ e 2 s,
we obtain J = Z Ω fu1− de1 ρ− 1 ˜ ρ? (?du ∧ ?e 1 u) + e 1 s∧ 1 ˜ ρd˜s ∧ e2 u +(fρ1− de1 u) ∧ e 2 ρ+ fs1−1 ˜ ρd˜s ∧ e 1 u ∧ e2 s = 0 (6.9)
for all e2ρ ∈ Λ0(Ω), e2u ∈ Λ2(Ω), e2s ∈ Λ0(Ω) for which tr(e2ρ) = 0 and tr(e2u) = 0. Finally, (6.9) can only be satisfied if
fρ1= de1u, (6.10) fu1= de1ρ+1 ˜ ρ? (?du ∧ ?e 1 u) − e 1 s∧ 1 ˜ ρd˜s, (6.11) fs1= 1 ˜ ρd˜s ∧ e 1 u, (6.12)
which are the conditions stated in the definition of D in (6.4). We still need to verify the boundary conditions. Insert (6.10),(6.11) and (6.12) into (6.8) and obtain
J = Z Ω e1ρ∧ de2 u+ e 2 ρ∧ de 1 u+ e 1 u∧ de 2 ρ+ e 1 u∧ T 2+ e1 u∧ S 2+ e2 u∧ de 1 ρ+ e 2 u∧ T 1 +e2u∧ (−e1s∧ 1 ˜ ρd˜s) + e 1 s∧ 1 ˜ ρd˜s ∧ e 2 u+ e 2 s∧ 1 ˜ ρd˜s ∧ e 1 u + Z ∂Ω (e1b∧ tr(e2 ρ) − tr(e 2 u) ∧ f 1 b) = 0, ∀e 2 ρ∈ Λ 0(Ω), e2 u∈ Λ 2(Ω), e2 s∈ Λ 0(Ω).
Using again the Leibniz rule for the underlined terms, we obtain J = Z Ω d(e1 ρ∧ e 2 u) + d(e 1 u∧ e 2 ρ) + e 1 u∧ T 2+ e2 u∧ T 1 + Z ∂Ω (e1b∧ tr(e2 ρ) − tr(e 2 u) ∧ f 1 b) = 0.
Finally, applying Stokes’ theorem and using the skew-symmetric property of F (e1
u, e2u) yields J =
Z ∂Ω
tr(e2u)∧(tr(e1ρ)−fb1)+(tr(e1u)+e1b)∧tr(e2ρ) = 0, ∀e2 ρ∈ Λ
0(Ω), e2 u∈ Λ
2(Ω).
The integral J can only be zero if
fb1= tr(e1ρ) e1b = − tr(e1u),
6.2. Pseudo-Poisson bracket for the primitive variables derived from the Stokes-Dirac structure
Using the approach outlined in17, we can associate a pseudo-Poisson bracket to the Stokes-Dirac structure defined in (6.4). The resulting skew-symmetric bracket has the form {k1, k1}D= − Z Ω δρk1∧ d(δuk2) + δuk1∧ d(δρk2) + ?du ˜ ρ ∧ ?δuk 2 ∧ ?δuk1 +d˜s ˜ ρ ∧ δsk 1∧ δ uk2− δsk2∧ δuk1 + Z ∂Ω tr(δuk1) ∧ tr(δρk2), (6.13)
where k1, k2 belong to the set of functions
k : Λ3(Ω) × Λ1(Ω) × Λ3(Ω) × Λ0(∂Ω) → R whose derivatives
δk = (δρk, δuk, δsk, δbk) ∈ Λ0(Ω) × Λ0(Ω) × Λ0(Ω) × Λ2(∂Ω) satisfy
δbk = − tr(δuk).
Remark 6.1. Note that if we integrate by parts the second term in the bracket of Morrison and Green (5.1), we obtain the bracket (6.13) derived from the Stokes-Dirac structure. Moreover, this bracket is skew-symmetric. The Jacobi identity for {·, ·}Din (6.13) is not automatically satisfied, and we call therefore {·, ·}Da pseudo-Poisson bracket as in 17.
6.3. Stokes-Dirac structure for vorticity-dilatation formulation of the compressible Euler equations
In this section we determine the Stokes-Dirac structure for the non-isentropic com-pressible Euler equations when written in the density weighted vorticity and di-latation variables. The linear spaces on which the Stokes-Dirac structure for the ρ, ω, θ, s variables will be defined are:
F : = Λ3(Ω) × Λ2(Ω) × Λ0(Ω) × Λ3(Ω) × Λ0(∂Ω) (6.14) E : = Λ0(Ω) × Λ1(Ω) × Λ3(Ω) × Λ0(Ω) × Λ2(∂Ω). (6.15) The Stokes-Dirac structure for the new variables is defined in the following theorem. Since the steps of the proof are analogous to the ones in the proof of Theorem 6.1, we only give the main steps in Appendix B.
Theorem 6.2. Let Ω ⊂ R3 be a three dimensional manifold with boundary ∂Ω. Consider F and E as given in (6.14) and (6.15) respectively, together with the bilinear form (f1, e1), (f2, e2) D:= Z Ω (e1ρ∧ fρ2+ e 2 ρ∧ f 1 ρ + e 1 ω∧ f 2 ω+ e 2 ω∧ f 1 ω+ e 1 θ∧ f 2 θ + e 2 θ∧ f 1 θ + e 1 s∧ f 2 s+ e 2 s∧ f 1 s) + Z ∂Ω (e1b∧ fb2+ e 2 b∧ f 1 b), (6.16) where fi= (fρi, fωi, fθi, fsi, fbi) ∈ F , ei= (eiρ, eiω, eiθ, eis, eib) ∈ E , i = 1, 2. Then D ⊂ F × E defined as D = { ((fρ, fω, fθ, fs, fb), (eρ, eω, eθ, es, eb)) ∈ F × E | fρ= d( p ˜ ρ ∧ dσ(e)), fω= dγ(¯e), fθ= −δγ(¯e), fs= 1 √ ˜ ρ∧ d˜s ∧ σ(e), fb= tr eρ+ ? ζ 2 ˜ρ∧ σ(e) , eb= −tr p ˜ ρ ∧ σ(e), tr(eω) = 0, tr(?eθ) = 0}, (6.17)
with σ(e) = deω+ δeθ, ¯e = (eρ, eω, eθ, es), is a Stokes-Dirac structure with respect to the bilinear form , D defined in (6.16).
6.4. Pseudo-Poisson bracket for the vorticity-dilatation variable derived from the Stokes-Dirac structure
Using the approach outlined in17, we can define a pseudo-Poisson bracket from the Stokes-Dirac structure (6.17) as follows
{k1, k1} D= − Z Ω δρk1∧ d p ˜ ρ ∧ dα(k2)+√d˜s ˜ ρ∧ δsk 1∧ α(k2) +δωk1∧ d(γ(grad k2)) − δθk1∧ δ(γ(grad k2)) + Z ∂Ω trpρ ∧ α(k˜ 1)∧ tr δρk2+ ? ζ 2 ˜ρ∧ α(k 2) , (6.18) where α(k) = d(δωk) + δ(δθk) and the operator γ(·) is defined in (5.16). Here k1, k2 belong to the set of functions
k : Λ3(Ω) × Λ2(Ω) × Λ0(Ω) × Λ3(Ω) × Λ0(∂Ω) → R whose derivatives
satisfy (see last two lines of (6.18)) δbk = − tr(
p ˜
ρ ∧ α(k)), tr(δωk) = 0, tr(?δθk) = 0.
Remark 6.2. Note that the bracket defined in (5.14) is identical to the one obtained from the Stokes-Dirac structure (6.18). Hence, this bracket is skew-symmetric.
6.5. Distributed-parameter port-Hamiltonian system
In this last section we make the connection between the Hamiltonian system and Stokes-Dirac structure for the non-isentropic compressible Euler equations in vorticity-dilatation formulation. Consider the Hamiltonian density H in (4.13) and total energy H in (4.14), with the gradient vector denoted as
grad H = (δρH, δωH, δθH, δsH) ∈ H∗Λ0× B∗1× B
3× H∗Λ0. Consider now the time function
F : R → HΛ3× B2× B∗0× HΛ 3, t 7→ (ρ(t), ω(t), θ(t), s(t))
and the Hamiltonian H[ρ(t), ω(t), θ(t), s(t)] along this trajectory. Then, dH dt = Z Ω δρH ∧ ∂ρ ∂t + δωH ∧ ∂ω ∂t + δθH ∧ ∂θ ∂t + δsH ∧ ∂s ∂t.
The variables ρt, ωt, θtand strepresent generalized velocities of the energy variables ρ, ω, θ, s. They are connected to the Stokes-Dirac structure D in (6.17) by setting
fρ= − ∂ρ ∂t, fω= − ∂ω ∂t, fθ= − ∂θ ∂t, fs= − ∂s ∂t.
Finally, setting eρ = δρH, eω = δωH, eθ = δθH, es = δsH, in the Stokes-Dirac structure, we obtain the distributed-parameter port-Hamiltonian system for the non-isentropic compressible Euler equations.
In order to simplify notations, let us split the operator γ(grad G) in (5.16) acting on a functional G, as follows
γ(grad G) = γρ(grad G) + γω,θ(grad G) + γs(grad G), (6.19) where γω,θ(grad G) = ζ 2 ˜ρ∧ ?d( p ˜ ρ ∧ α(G)) +pρ ∧ d˜ ?(α(G) ∧ ζ 2 ˜ρ) + ? ?d ζ √ ˜ ρ ∧ ?α(G) , (6.20)
with α(G) defined in (5.5), and γρ(grad G) = p ˜ ρ ∧ dδG δρ, γs(grad G) = − 1 √ ˜ ρ∧ d˜s ∧ δG δs.
Corollary 6.1. The distributed-parameter port-Hamiltonian system for the three dimensional manifold Ω, state space HΛ3× B2× B∗
0× HΛ3, Stokes-Dirac structure D, (6.17), and Hamiltonian H, (4.13), is given as
−∂ρ∂t −∂ω ∂t −∂θ ∂t −∂s ∂t = 0 d(√ρ ∧ d·) d(˜ √ρ ∧ δ·)˜ 0 dγρ(·) dγω(·) dγθ(·) dγs(·) −δγρ(·) −δγω(·) −δγθ(·) −δγs(·) 0 √d˜s ˜ ρ∧ d· d˜s √ ˜ ρ∧ δ· 0 δρH δωH δθH δsH "fb eb # = tr 1 ?2 ˜ζρ∧ d· ?2 ˜ζρ ∧ δ· 0 −√ρ ∧ d·˜ −√ρ ∧ δ·˜ δρH δωH δθH . (6.21)
Note that (6.21) might be a good starting point for nonlinear boundary control systems. By the power-conserving property of any Stokes-Dirac structure, i.e.,
(f, e), (f, e) D= 0, ∀(f, e) ∈ D,
it follows that any distributed-parameter port-Hamiltonian system satisfies along its trajectories the energy balance
dH dt =
Z ∂Ω
eb∧ fb. (6.22)
This expresses that the increase in internally stored energy in the domain Ω is equal to the power supplied to the system through the boundary ∂Ω.
7. Conclusions
The main results of this article concern a novel Hamiltonian vorticity-dilatation formulation of the compressible Euler equations. This formulation uses the den-sity weighted vorticity and dilatation, together with the entropy and denden-sity, as primary variables. We obtained this new formulation using the following steps. First, we defined the Hamiltonian functional with respect to the chosen primary variables and calculated its functional derivatives. Next, we derived a pseudo-Poisson bracket, (port)-Hamiltonian formulation and Stokes-Dirac structure for the vorticity-dilatation formulation of the compressible Euler equations and showed the relation between these different formulations. An essential tool in this analysis was the use of the Hodge decomposition on bounded domains. These results extend the vorticity-streamfunction formulation of the Euler equations for incompressible flows to compressible flows.
The long term goal of this research is the development of finite element formula-tions that preserve these mathematical structures also at the discrete level. A nice direction for port-Hamiltonian systems using mixed finite elements is described in 8 or using pseudo-spectral methods in 14,?. In future research we will explore this
using the concept of discrete differential forms and exterior calculus as outlined in 2,3.
Appendix A. Proof of Lemma 5.2
In this appendix we give the proof of Lemma 5.2.
Proof. Using the notations above, we obtain for the T1-term in (5.1) T1= − Z Ω δF δρ ∧ d p ˜ ρ ∧ α(G)+ α(F ) ∧ ζ 2 ˜ρ∧ ?d p ˜ ρ ∧ α(G) − δG δρ ∧ d p ˜ ρ ∧ α(F )− α(G) ∧ ζ 2 ˜ρ∧ ?d p ˜ ρ ∧ α(F ) . (A.1) Using the integration by parts formula (2.9), we rewrite the last two terms in (A.1) as follows ?δG δρ + α(G) ∧ ζ 2 ˜ρ, d p ˜ ρ ∧ α(F ) = δ ?δG δρ + α(G) ∧ ζ 2 ˜ρ ,pρ ∧ α(F )˜ + Z ∂Ω tr(pρ ∧ α(F )) ∧ tr˜ δG δρ + ?(α(G) ∧ ζ 2 ˜ρ) = Z Ω d δG δρ + ?(α(G) ∧ ζ 2 ˜ρ) ∧pρ ∧ α(F )˜ + Z ∂Ω tr(pρ ∧ α(F )) ∧ tr˜ δG δρ + ?(α(G) ∧ ζ 2 ˜ρ) . (A.2)
Next, we consider the T2-term and introduce the new variables ω and θ into (5.1) to obtain ? ?δG δu ∧ ? δF δu = ˜ρ ∧ ? (?α(G) ∧ ?α(F )) , and using (3.1), T2= − Z Ω ? iX ( ˜ρ ∧ ? (?α(G) ∧ ?α(F ))) = − Z Ω ?du ∧ ?α(G) ∧ ?α(F ),
with X = (?duρ˜ )]. Similarly, the term T
3 in (5.1) can be transformed into the new variables as T3= − Z Ω d˜s √ ˜ ρ∧ δF δs ∧ α(G) − δG δs ∧ α(F ) = − Z Ω δF δs ∧ d˜s √ ˜ ρ∧ α(G) − δG δs ∧ d˜s √ ˜ ρ∧ α(F ).
Adding all terms, the bracket (5.1) in terms of the variables ρ, ω, θ, s has the form {F , G} = − Z Ω δF δρ ∧ d p ˜ ρ ∧ α(G)+δF δs ∧ 1 √ ˜ ρ∧ d˜s ∧ α(G) + α(F ) ∧ γ(grad G) + Z ∂Ω tr(pρ ∧ α(F )) ∧ tr˜ δG δρ + ?(α(G) ∧ ζ 2 ˜ρ) , (A.3)
where γ(grad G) is given in (5.16). In the following we expand the last integral over the domain Ω in (A.3) as
hα(F ), ?γ(grad G)i = Z Ω δF δω ∧ dγ(grad G) − δF δθ ∧ δγ(grad G) + Z ∂Ω tr(δF δω) ∧ tr(γ(grad G)) − tr(? δF δθ) ∧ tr(?γ(grad G)) , If we use the boundary assumptions (5.2) and (5.3) for the variational derivatives, the last boundary integral cancels and we obtain (5.14).
Appendix B. Proof of Theorem 6.2
In this appendix we show the main steps of the proof of Theorem 6.2. Proof. The proof of Theorem 6.2 consists of two steps.
Step 1. First we show that D ⊂ D⊥. Let (f1, e1) ∈ D fix, and consider any (f2, e2) ∈ D. Substituting the definition of D into (6.16), we obtain that
I : = (f1, e1), (f2, e2) D = Z Ω h e1ρ∧ d(pρ ∧ σ(e˜ 2)) + e2ρ∧ d(pρ ∧ σ(e˜ 1)) +e1s∧√d˜s ˜ ρ∧ σ(e 2) + e2 s∧ d˜s √ ˜ ρ∧ σ(e 1)i + Z Ω e1 ω∧ dγ(¯e 2) + e2 ω∧ dγ(¯e 1) − e1 θ∧ δγ(¯e 2) − e2 θ∧ δγ(¯e 1) + Z ∂Ω e1b∧ tr e2ρ+ ? ζ 2 ˜ρ∧ σ(e 2) + e2b∧ tr e1ρ+ ? ζ 2 ˜ρ∧ σ(e 1) . Consider I1= Z Ω h e1ρ∧ d(pρ ∧ (de˜ 2ω+ δe2θ)) + eω2 ∧ dγρ(¯e1) − e2θ∧ δγρ(¯e1) i
and apply the integration by parts formula (2.9) for the underlined terms, to obtain ?e2 ω, dγρ(¯e1) = δ ? e2ω, γρ(¯e1) + Z ∂Ω tr(γρ(¯e1)) ∧ tr(e2ω) = Z Ω de2ω∧ γρ(¯e1)
and ?e2 θ, δγρ(¯e1) = d ? e2θ, γρ(¯e1) − Z ∂Ω tr(?γρ(¯e1)) ∧ tr(?e2θ) = − Z Ω δe2θ∧ γρ(¯e1), where we used that tr(e2
ω) = 0 and tr(?e2θ) = 0. Inserting the definition of γρ(¯e1), we obtain that I1= Z Ω h e1ρ∧ d(p ˜ ρ ∧ σ(e2)) + de2ω∧ γρ(¯e1) + δe2θ∧ γρ(¯e1) i = Z Ω dpρ ∧ σ(e˜ 2) ∧ e1ρ= Z ∂Ω trpρ ∧ σ(e˜ 2)∧ tr(e1 ρ). Similarly, we obtain that
I2= Z Ω h e2ρ∧ d(p ˜ ρ ∧ σ(e1)) + e1ω∧ dγρ(¯e2) − e1θ∧ δγρ(¯e2) i = Z ∂Ω trpρ ∧ σ(e˜ 1)∧ tr(e2 ρ). Next, let I3= Z Ω e1 ω∧ d(γω,θ(¯e2)) + e2ω∧ d(γω,θ(¯e1)) − e1θ∧ δ(γω,θ(¯e2)) − e2θ∧ δ(γω,θ(¯e1)) , with γω,θ(·) defined in (6.20). Note that when applied to ¯ei, α(F ) is replaced by σ(ei). Apply again partial integration and use that tr(ei
ω) = 0 and tr(?eiθ) = 0 to obtain I3= Z Ω de1 ω∧ γω,θ(¯e2) + de2ω∧ γω,θ(¯e1) + δe1θ∧ γω,θ(¯e2) + δe2θ∧ γω,θ(¯e1) = Z Ω σ(e1) ∧ γ ω,θ(¯e2) + σ(e2) ∧ γω,θ(¯e1) .
Inserting the definition of γω,θ(¯ei), i = 1, 2 and applying again partial integration, the above integral will reduce to
I3= Z ∂Ω tr(pρ∧σ(e˜ 2))∧tr ?(σ(e1) ∧ ζ 2 ˜ρ) +tr(pρ∧σ(e˜ 1))∧tr ?(σ(e2) ∧ ζ 2 ˜ρ) . Finally, observe that the term containing the entropy
I4= Z Ω e1s∧ d˜s √ ˜ ρ∧ σ(e 2 ) + e2s∧ d˜s √ ˜ ρ∧ σ(e 1 ) + σ(e2) ∧ γs(¯e1) + σ(e1) ∧ γs(¯e2) , is zero when we insert γs(¯ei) = −√d˜sρ˜∧ eis, i = 1, 2. Combining all terms, we obtain that I = Z ∂Ω tr e1ρ+ ? σ(e1) ∧ ζ 2 ˜ρ ∧tr(pρ ∧ σ(e˜ 2)) + e2b + tr e2ρ+ ? σ(e2) ∧ ζ 2 ˜ρ ∧tr(pρ ∧ σ(e˜ 1)) + e1b= 0,
